The hom-associative Weyl algebras in prime characteristic

TL;DR

This paper introduces and studies the properties of hom-associative Weyl algebras in prime characteristic, generalizing classical Weyl algebras and exploring their algebraic structure and deformations.

Contribution

It defines the first hom-associative Weyl algebras in prime characteristic, analyzes their structural properties, and establishes their deformation relationships with classical Weyl algebras.

Findings

All nonzero endomorphisms are injective but not surjective.

The algebras can be described as multi-parameter formal hom-associative deformations.

They induce multi-parameter formal hom-Lie deformations of the associated Lie algebra.

Abstract

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general "twisting" procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, we show that they can be described as a multi-parameter formal hom-associative deformation of the first associative Weyl algebra, and that this deformation induces a multi-parameter formal hom-Lie deformation of the corresponding Lie algebra, when using the commutator as bracket.